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Spectral Stability of Inviscid Roll Waves

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Abstract

We carry out a systematic analytical and numerical study of spectral stability of discontinuous roll wave solutions of the inviscid Saint-Venant equations, based on a periodic Evans–Lopatinsky determinant analogous to the periodic Evans function of Gardner in the (smooth) viscous case, obtaining a complete spectral stability diagram useful in hydraulic engineering and related applications. In particular, we obtain an explicit low-frequency stability boundary, which, moreover, matches closely with its (numerically-determined) counterpart in the viscous case. This is seen to be related to but not implied by the associated formal first-order Whitham modulation equations.

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Acknowledgements

Thanks to Olivier Lafitte for stimulating discussions regarding normal forms for singular ODE, and to Blake Barker for his generous help in sharing source computations from [BJN+17]. The numerical computations in this paperwere carried out in the MATLAB environment; analytical calculations were checked with the aid of MATLAB’s symbolic processor. Thanks to University Information Technology Services (UITS) division from Indiana University for providing the Karst supercomputer environment in which most of our computations were carried out. This research was supported in part by Lilly Endowment, Inc., through its support for the Indiana University Pervasive Technology Institute, and in part by the Indiana METACyt Initiative. The Indiana METACyt Initiative at IU was also supported in part by Lilly Endowment, Inc.

Author information

Authors and Affiliations

  1. University of Kansas, Lawrence, KS, 66045, USA

    Mathew A. Johnson

  2. Institut deMathématiques de Toulouse, UMR5219, CNRS, INSA, Université de Toulouse, 31077, Toulouse, France

    Pascal Noble

  3. Univ Rennes, CNRS, IRMAR - UMR 6625, 35000, Rennes, France

    L. Miguel Rodrigues

  4. Indiana University, Bloomington, IN, 47405, USA

    Zhao Yang & Kevin Zumbrun

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  1. Mathew A. Johnson
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  2. Pascal Noble
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  3. L. Miguel Rodrigues
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  4. Zhao Yang
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  5. Kevin Zumbrun
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Correspondence to Mathew A. Johnson.

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Communicated by W. Schlag

Research of M.J. was partially supported under NSF Grant No. DMS-1614785. Research of P.N. was partially supported by the French ANR Project BoND ANR-13-BS01-0009-01. Research of M.R.was partially supported by the French ANR Project BoND ANR-13-BS01-0009-01 and the city of Rennes. Research of Z.Y. was partially supported by the Hazel King Thompson Summer Reading Fellowship. Research of K.Z. was partially supported under NSF Grant No. DMS-1400555 and DMS-1700279.

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Johnson, M.A., Noble, P., Rodrigues, L.M. et al. Spectral Stability of Inviscid Roll Waves. Commun. Math. Phys. 367, 265–316 (2019). https://doi.org/10.1007/s00220-018-3277-7

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